Orthogonal transform apparatus, orthogonal transform method, orthogonal transform computer program, and audio decoding apparatus

ABSTRACT

An orthogonal transform apparatus computes either one of the real and imaginary components of the quadrature mirror filter coefficient contained in a first subinterval of a plurality of subintervals among which a coefficient sequence containing a plurality of quadrature mirror filter coefficients is divided so that the values of basis functions are symmetrically placed, by computing a sum of products of the plurality of modified discrete cosine transform coefficients and the basis functions corresponding to the subinterval, computes the other one of the real and imaginary components of the quadrature mirror filter coefficient contained in the first subinterval and the real and imaginary components of the quadrature mirror filter coefficient contained in another subintervals by performing a butterfly operation using a computed value produced as a result of the sum of products and computes each quadrature mirror filter coefficient by combining the real component and imaginary component thereof.

CROSS-REFERENCE TO RELATED APPLICATION

This application is based upon and claims the benefit of priority of theprior Japanese Patent Application No. 2013-070436, filed on Mar. 28,2013, the entire contents of which are incorporated herein by reference.

FIELD

The embodiments discussed herein are related to an orthogonal transformapparatus, an orthogonal transform method, an orthogonal transformcomputer program, and an audio decoding apparatus using the same.

BACKGROUND

Various audio coding systems have been developed in the prior art forcompressing the amount of data needed to represent a multichannel audiosignal carrying three or more channels. One such known coding system isthe MPEG Surround System standardized by the Moving Picture ExpertsGroup (MPEG) (for example, refer to ISO/IEC 23003-1). In the MPEGSurround System, a plurality of channel signals are downmixed togenerate spatial information and a main signal representing the maincomponent of each original channel signal, and this main signal and thespatial information are encoded. Further, in this coding system, aresidual signal representing a component orthogonal to the main signalis also computed, and this residual signal may also be encoded.

The main signal and the residual signal are each obtained by firsttransforming the downmix signal into a time domain signal and thentransforming it into a frequency domain signal by a modified discretecosine transform (MDCT). Of these two signals, the main signal oncetransformed into the time domain signal is further transformed into QMFcoefficients representing a time-frequency domain signal by using aquadrature mirror filter (QMF), because upmixing is performed using thespatial information when decoding. Therefore, the residual signal in thefrequency domain is also transformed into QMF coefficients in thetime-frequency domain so that the residual signal can be used whenupmixing.

SUMMARY

Since the orthogonal transforms such as MDCT and QMF are performed overand over again as described above to decode the audio signal encoded bythe MPEG Surround System, the amount of computation becomes very large.The larger the amount of computation, the higher is the computationalcapability demanded of audio decoding apparatus, and the powerconsumption of the audio decoding apparatus correspondingly increases.There is thus a need to reduce the amount of computation needed todecode the audio signal encoded by the MPEG Surround System.

According to one embodiment, an orthogonal transform apparatus fortransforming a plurality of modified discrete cosine transformcoefficients contained in a prescribed interval into a coefficientsequence containing a plurality of quadrature mirror filter coefficientsis provided. The orthogonal transform apparatus includes: an inverseexponential transform unit which computes either one of the real andimaginary components of the quadrature mirror filter coefficientcontained in a first subinterval of a plurality subintervals among whichthe coefficient sequence is divided so that the values of basisfunctions used to compute the coefficient sequence are symmetricallyplaced, by computing a sum of products of the plurality of modifieddiscrete cosine transform coefficients and the basis functionscorresponding to the first subinterval, and computes the other one ofthe real and imaginary components of the quadrature mirror filtercoefficient contained in the first subinterval and the real andimaginary components of the quadrature mirror filter coefficientcontained in another subintervals of the plurality of subintervals byperforming a butterfly operation using a computed value produced as aresult of the sum of products; and a coefficient adjusting unit whichcomputes the quadrature mirror filter coefficients by combining the realcomponent and the imaginary component for each of the plurality ofquadrature mirror filter coefficients.

The object and advantages of the invention will be realized and attainedby means of the elements and combinations particularly pointed out inthe claims.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory and arenot restrictive of the invention, as claimed.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a diagram schematically illustrating the configuration of anaudio decoding apparatus incorporating an orthogonal transform apparatusaccording to one embodiment.

FIG. 2 is a diagram illustrating one example of a data format carryingan encoded audio signal.

FIG. 3 is a diagram illustrating one example of a quantization tablethat provides a mapping of degrees of similarity.

FIG. 4 is a diagram illustrating one example of a quantization tablethat provides a mapping of intensity differences.

FIG. 5 is a diagram illustrating one example of a quantization tablethat provides a mapping of prediction coefficients.

FIG. 6 is a conceptual diagram for explaining how MDCT coefficients aretransformed into QMF coefficients.

FIG. 7 is a block diagram of an orthogonal transform unit.

FIG. 8 is a diagram for explaining the periodicity of basis functions.

FIG. 9 is a diagram for explaining the difference between an IMDCT basisfunction and an IMDST basis function.

FIG. 10 is a diagram for explaining the processing performed by thelocal multiplication unit.

FIG. 11 is a diagram for explaining one example of a butterfly operationperformed by a coefficient computing unit.

FIG. 12 is a diagram for explaining another example of a butterflyoperation performed by the coefficient computing unit.

FIG. 13 is an operation flowchart of an orthogonal transform process.

FIG. 14 is an operation flowchart of an audio decoding process which isperformed by the audio decoding apparatus.

FIG. 15 is a block diagram of an orthogonal transform apparatusaccording to a second embodiment.

FIG. 16 is an operation flowchart of a switching process.

FIG. 17 is a block diagram of a second inverse modified discreteexponential transform unit.

FIG. 18 is a diagram illustrating the relationship between the cosinebasis function of a butterfly IMDCT and the cosine basis function of aconventional IMDCT.

FIG. 19 is a diagram for explaining how the MDCT coefficients arereordered within a butterfly computation interval.

FIG. 20 is a block diagram of an inverse cosine transform unit.

FIG. 21 is a diagram of graphs illustrating the relationship between thenumber, M, of MDCT coefficients contained in a computation interval andthe amount of computation for IMDET that utilizes the symmetry of basisfunctions in comparison with that for IMDET that uses FFT.

FIG. 22 is a diagram illustrating the configuration of a computer thatoperates as the audio decoding apparatus by executing a computer programfor implementing the functions of the various units constituting theaudio decoding apparatus according to the above embodiment or itsmodified example.

DESCRIPTION OF EMBODIMENTS

An orthogonal transform apparatus according to one embodiment will bedescribed below with reference to the drawings. In the process fordecoding the audio signal encoded by the MPEG Surround System, theprocess of transforming the frequency domain residual signal (MDCTcoefficients) into the time-frequency domain signal (QMF coefficients)requires the largest amount of computation. For example, in the ISOreference decoder, the amount of computation involved in thistransforming process accounts for about 70% of the total amount ofcomputation of the decoding process. According, if the amount ofcomputation needed to transform the MDCT coefficients into the QMFcoefficients can be reduced, it becomes possible to reduce the amount ofcomputation needed to decode the audio signal encoded by the MPEGSurround System.

In view of the above, the orthogonal transform apparatus of theembodiment aims to reduce the amount of computation needed to transformthe MDCT coefficients into the QMF coefficients. For this purpose, theorthogonal transform apparatus exploits the symmetry of basis functionsin a butterfly inverse modified discrete cosine transform (IMDCT) and abutterfly inverse modified discrete sine transform (IMDST) that are usedwhen transforming the MDCT coefficients into the QMF coefficients. Inthis patent specification, IMDCT and IMDST are collectively referred toas the inverse modified discrete exponential transform (IMDET).

In the present embodiment, the multichannel audio signal to be decodedis a 5.1-channel audio signal. However, the multichannel audio signal tobe decoded is not limited to a 5.1-channel audio signal, but may be, forexample, a 7.1-channel audio signal.

FIG. 1 is a diagram schematically illustrating the configuration of anaudio decoding apparatus 1 according to one embodiment. The audiodecoding apparatus 1 includes a demultiplexing unit 11, a main signaldecoding unit 12, a time-frequency transform unit 13, a spatialinformation decoding unit 14, a residual signal decoding unit 15, anorthogonal transform unit 16, an upmixing unit 17, and a frequency-timetransform unit 18.

These units constituting the audio decoding apparatus 1 are eachimplemented as a separate circuit. Alternatively, these unitsconstituting the audio decoding apparatus 1 may be implemented on theaudio decoding apparatus 1 in the form of a single integrated circuit onwhich the circuits corresponding to the respective units are integrated.Further alternatively, these units constituting the audio decodingapparatus 1 may be implemented as functional modules by executing acomputer program on a processor contained in the audio decodingapparatus 1.

The demultiplexing unit 11 demultiplexes a main signal code, a spatialinformation code, and an encoded residual signal from a data streamcontaining the encoded audio signal in accordance with a data formatcarrying the encoded audio signal. The main signal code includes anAdvanced Audio Coding (AAC) code and a Spectral Band Replication (SBR)code.

FIG. 2 is a diagram illustrating one example of the data format carryingthe encoded audio signal. In the illustrated example, the encoded audiosignal is created in accordance with the MPEG-4 ADTS (Audio DataTransport Stream) format. In the encoded data sequence 200 depicted inFIG. 2, the AAC code is stored in a data block 210. The SBR code, thespatial information code, and the encoded residual signal are stored indesignated areas within a data block 220 that stores FILL elements ofthe ADTS format.

The demultiplexing unit 11 passes the main signal code to the mainsignal decoding unit 12. Further, the demultiplexing unit 11 passes thespatial information code to the spatial information decoding unit 14 andthe encoded residual signal to the residual signal decoding unit 15.

The main signal decoding unit 12 decodes the main signal code which isan encoded version of the main signal representing the main component ofa stereo signal generated by downmixing the original multichannel audiosignal. The main signal decoding unit 12 reconstructs low-frequencycomponents for the left and right channels by decoding the AAC code inaccordance with, for example, the AAC code decoding process described inthe ISO/IEC 14496-3 specification. More specifically, the main signaldecoding unit 12 reconstructs a quantized signal by entropy decoding theAAC code, and reconstructs the MDCT coefficients by inverse-quantizingthe quantized signal. Then, the main signal decoding unit 12reconstructs the low-frequency components for the left and rightchannels on a frame-by-frame basis by applying the IMDCT to thereconstructed MDCT coefficients.

Further, the main signal decoding unit 12 reconstructs high-frequencycomponents for the left and right channels on a frame-by-frame basis bydecoding the SBR code in accordance with, for example, the SBR codedecoding process described in the ISO/IEC 14496-3 specification. Then,the main signal decoding unit 12 reconstructs the left and right channelsignals of the stereo signal by combining the low-frequency componentsand high-frequency components on a channel-by-channel basis. The mainsignal decoding unit 12 passes the reconstructed stereo signal to thetime-frequency transform unit 13.

The time-frequency transform unit 13 is one example of a quadraturemirror filtering unit, and transforms each of the time-domain channelsignals of the reconstructed stereo signal into the QMF coefficients inthe time-frequency domain on a frame-by-frame basis by using a QMFfilter bank.

The QMF filter bank is expressed by the following equation.

$\begin{matrix}{{{{QMF}\left( {k,n} \right)} = {\exp\left\lbrack {j\frac{\pi}{128}\left( {k + 0.5} \right)\left( {{2\; n} + 1} \right)} \right\rbrack}},{0 \leq k < 64},{0 \leq n < 128}} & (1)\end{matrix}$where n is a variable representing the time, and represents the nth timewhen the stereo signal of one frame is divided into 128 equal partsalong its time direction. The frame length can be set to any value thatfalls within the range of 10 to 80 msec. On the other hand, k is avariable representing the frequency band, and represents the kthfrequency band when the frequency band of the frequency signal isdivided into 64 equal parts.

The time-frequency transform unit 13 passes the computed QMFcoefficients to the upmixing unit 17.

The spatial information decoding unit 14 decodes the spatial informationcode received from the demultiplexing unit 11. The spatial informationincludes, for example, the degree of similarity ICC between two channelsthat represents the degree of sound spreading, and the intensitydifference CLD between two channels that represents the degree of soundlocalization. The spatial information further includes a predictioncoefficient CPC for predicting the center channel signal from the rightand left channel signals. The degree of similarity ICC, the intensitydifference CLD, and the prediction coefficient CPC are obtained on afrequency-by-frequency basis when downmixing the audio signal. Thespatial information code includes a Huffman code for each of the degreeof similarity ICC, the intensity difference CLD, and the predictioncoefficient CPC.

The spatial information decoding unit 14 reconstructs an indexdifference value by referring to a table that provides a mapping betweenthe index difference value, such as the degree of similarity ICC betweenadjacent frequencies, and the Huffman code. The spatial informationdecoding unit 14 reconstructs the index value for each frequency band bysequentially adding up the index differences on a frequency-band byfrequency-band basis. Then, the spatial information decoding unit 14determines the quantized value representing the degree of similarityICC, the intensity difference CLD, or the prediction coefficient CPCcorresponding to that index value, by referring to a table that providesa mapping between the index value and the quantized value representingthe degree of similarity ICC, the intensity difference CLD, or theprediction coefficient CPC, respectively.

FIG. 3 is a diagram illustrating one example of a quantization table forthe degree of similarity. In the quantization table 300 illustrated inFIG. 3, each entry in the upper row 310 carries the index value, andeach entry in the lower row 320 carries the quantized value for thedegree of similarity corresponding to the index value in the samecolumn. The range of values that the degree of similarity can take isfrom −0.99 to +1. For example, when the index value of the frequencyband k is 3, the spatial information decoding unit 14 refers to thequantization table 300 and determines that 0.60092 corresponding to theindex value 3 is the quantized value of the degree of similarity.

FIG. 4 is a diagram illustrating one example of a quantization table forthe intensity difference. In the quantization table 400 illustrated inFIG. 4, each entry in rows 410, 430, and 450 carries the index value,and each entry in rows 420, 440, and 460 carries the quantized value forthe intensity difference corresponding to the index value in the samecolumn in the corresponding one of the rows 410, 430, and 450. Forexample, when the index value of the frequency band k is 5, the spatialinformation decoding unit 14 refers to the quantization table 400 anddetermines that 10 corresponding to the index value 5 is the quantizedvalue of the density difference.

FIG. 5 is a diagram illustrating one example of a quantization table forthe prediction coefficient. In the quantization table 500 illustrated inFIG. 5, each entry in rows 510, 520, 530, 540, and 550 carries the indexvalue, and each entry in rows 515, 525, 535, 545, and 555 carries thequantized value for the prediction coefficient corresponding to theindex value in the same column in the corresponding one of the rows 510,520, 530, 540, and 550. For example, when the index value of thefrequency band k is 3, the spatial information decoding unit 14 refersto the quantization table 500 and determines that 0.3 corresponding tothe index value 3 is the quantized value of the prediction coefficient.The spatial information decoding unit 14 passes the quantized values ofthe spatial information for each frequency band to the upmixing unit 17.

The residual signal decoding unit 15 decodes the encoded residual signalwhich is a component orthogonal to the main signal. In the MPEG SurroundSystem, since the residual signal is also AAC encoded, MDCT is appliedto the residual signal when encoding it. Accordingly, the residualsignal decoding unit 15 reconstructs the residual signal represented bythe MDCT coefficients, by decoding the residual signal in accordancewith, for example, the AAC code decoding method described in the ISO/IEC13818-7 specification. The MDCT coefficients are supplied to theorthogonal transform unit 16.

The orthogonal transform unit 16 is one example of the orthogonaltransform apparatus, and transforms the residual signal represented bythe MDCT coefficients as frequency domain signals into the QMFcoefficients as time-frequency domain signals. The details of theorthogonal transform unit 16 will be described later.

The upmixing unit 17 reconstructs the QMF coefficients for each channelof the 5.1-channel audio signal by upmixing, based on the spatialinformation, the QMF coefficients of the left and right channels of thestereo signal and the QMF coefficients of the residual signal for eachfrequency band. For this purpose, the upmixing unit 17 may use the upmixtechnique specified, for example, in the ISO/IEC 23003-1 specification.For example, by upmixing the QMF coefficients of the left and rightchannels of the stereo signal and the QMF coefficients of the residualsignal by using the spatial information, the upmixing unit 17reconstructs the QMF coefficients for three channels, i.e., the left,right, and center channels. Further, by upmixing the reconstructed leftchannel QMF coefficients by using the spatial information computed whendownmixing the front-left channel and the rear-left channel, theupmixing unit 17 reconstructs the QMF coefficients for the front-leftchannel and the rear-left channel. Similarly, by upmixing thereconstructed right channel QMF coefficients by using the spatialinformation computed when downmixing the front-right channel and therear-right channel, the upmixing unit 17 reconstructs the QMFcoefficients for the front-right channel and the rear-right channel.Further, by upmixing the reconstructed center channel QMF coefficientsby using the spatial information computed when downmixing the centerchannel and the bass channel, the upmixing unit 17 reconstructs the QMFcoefficients for the center channel and the bass channel.

The upmixing unit 17 passes the QMF coefficients reconstructed for eachchannel to the frequency-time transform unit 18.

The frequency-time transform unit 18 is one example of an inversequadrature mirror filtering unit, and reconstructs the 5.1-channel audiosignal by processing the QMF coefficients of each channel by reversingthe QMF filter bank process performed by the time-frequency transformunit 13. The frequency-time transform unit 18 outputs the reconstructedaudio signal, for example, to a speaker.

The orthogonal transform unit 16 will be described in detail below. FIG.6 is a conceptual diagram for explaining how the MDCT coefficients aretransformed into the QMF coefficients. The array of MDCT coefficients isconstructed from a plurality of coefficients arrayed only in thedirection of the frequency axis. On the other hand, the array of QMFcoefficients is constructed from a plurality of coefficients arrayed inthe direction of the time axis as well as in the direction of thefrequency axis.

In view of the above, when transforming the MDCT coefficients into theQMF coefficients, the orthogonal transform unit 16 divides the wholearray of the MDCT coefficients into a plurality of frequency bands, eachoverlapping with an adjacent frequency band by one half of the frequencyband, such as the frequency bands 603 to 605, in accordance with theISO/IEC 23003-1 specification. In this case, each frequency band has alength twice that of the frequency band to which the conventional IMDCTis applied, and contains, for example, a number, 2N, of successive MDCTcoefficients. Then, by applying the butterfly IMDET to each suchfrequency band, and thus canceling out aliasing distortions occurringbetween the frequency bands, the orthogonal transform unit 16 obtains anumber, 2N, of QMF coefficients arranged along the direction of the timeaxis for each frequency band. However, the amount of computationinvolved in the butterfly IMDET is very large. In view of this, theorthogonal transform unit 16 according to the present embodimentexploits the symmetry of the basis functions of the IMDET in order toreduce the amount of computation involved in the butterfly IMDET.

FIG. 7 is a block diagram of the orthogonal transform unit 16. Theorthogonal transform unit 16 includes a windowing unit 21, an inversemodified discrete exponential transform unit 22, and a coefficientadjusting unit 23.

The windowing unit 21 multiplies the MDCT coefficients of the residualsignal by a windowing function for the butterfly IMDCT and butterflyIMDST and a gain (½N)^(1/2). In the present embodiment, the windowingfunction w_(f)[n] is expressed by the following equation.

$\begin{matrix}{{{w_{f}\lbrack n\rbrack} = {\sum\limits_{m = {- 319}}^{319}\;{{h_{norm}\left\lbrack {320 + m} \right\rbrack}\cos\left\{ \frac{{\pi\left( {{2\; n} + 1 + {oddflag} - {2\; N}} \right)}m}{128\; N} \right\}}}}\mspace{20mu}{{h\lbrack n\rbrack} = \left\{ {{\begin{matrix}{h_{qmf}\lbrack n\rbrack} & {{{if}\mspace{14mu} 0} \leq n \leq 127} \\{- {h_{qmf}\lbrack n\rbrack}} & {{{if}\mspace{14mu} 128} \leq n \leq 255} \\{h_{qmf}\lbrack n\rbrack} & {{{if}\mspace{14mu} 256} \leq n \leq 383} \\{- {h_{qmf}\lbrack n\rbrack}} & {{{if}\mspace{14mu} 384} \leq n \leq 511} \\{h_{qmf}\lbrack n\rbrack} & {{{if}\mspace{14mu} 512} \leq n \leq 639}\end{matrix}.\mspace{20mu}{h_{norm}\lbrack n\rbrack}} = {{\frac{h\lbrack n\rbrack}{\sum\limits_{n = 0}^{639}\;{h\lbrack n\rbrack}}\mspace{20mu}{oddflag}} = \left\{ {\begin{matrix}1 & {{{if}\mspace{14mu}{N/2}\mspace{14mu}{is}\mspace{14mu}{odd}};} \\0 & {{if}\mspace{14mu}{N/2}\mspace{14mu}{is}\mspace{14mu}{even}}\end{matrix}.} \right.}} \right.}} & (2)\end{matrix}$where (2N) represents the number of MDCT coefficients contained in thefrequency band f to which the butterfly IMDET is applied. On the otherhand, n represents the order of the coefficient on the time axisobtained as a result of the butterfly IMDET. The windowing unit 21passes the MDCT coefficients of the residual signal multiplied by thewindowing function and the gain to the inverse modified discreteexponential transform unit 22.

For each of the plurality of intervals into which the entire frequencyband is divided, the inverse modified discrete exponential transformunit 22 performs the IMDET of the MDCT coefficients of the residualsignal multiplied by the windowing function and the gain, and therebycomputes the real and imaginary components of the QMF coefficients ofthe frequency corresponding to that interval.

For example, when the number of MDCT coefficients contained in eachinterval is 8, i.e., when N=4, the IMDET is expressed by the followingequation using a transform matrix.

$\begin{matrix}{{\begin{matrix}{\begin{matrix}{{QMF}\mspace{14mu}{coefficients}} \\\left( {{real}\mspace{14mu}{parts}} \right)\end{matrix}\left\{ \begin{matrix}\; \\\; \\\; \\\; \\\; \\\; \\\; \\\;\end{matrix} \right.} \\{\begin{matrix}{{QMF}\mspace{14mu}{coefficients}} \\\left( {{imaginary}\mspace{14mu}{parts}} \right)\end{matrix}\left\{ \begin{matrix}\; \\\; \\\; \\\; \\\; \\\; \\\; \\\;\end{matrix} \right.}\end{matrix}\begin{bmatrix}{y\; 1} \\{y\; 2} \\{y\; 3} \\{y\; 4} \\{y\; 5} \\{y\; 6} \\{y\; 7} \\{y\; 8} \\{y\; 9} \\{y\; 10} \\{y\; 11} \\{{y\; 12}\;} \\{y\; 13} \\{y\; 14} \\{y\; 15} \\{y\; 16}\end{bmatrix}} = {\quad{{{\begin{bmatrix}c_{0,0} & \ldots & c_{0,7} \\\vdots & \ddots & \vdots \\c_{7,0} & \ldots & c_{7,7} \\s_{0,0} & \ldots & s_{0,7} \\\vdots & \ddots & \vdots \\s_{7,0} & \ldots & s_{7,7}\end{bmatrix}\begin{bmatrix}{x\; 1} \\{x\; 2} \\{x\; 3} \\{x\; 4} \\{x\; 5} \\{x\; 6} \\{x\; 7} \\{x\; 8}\end{bmatrix}}\mspace{20mu} c_{n,k}} = {{{\cos\left\lbrack {\frac{\pi}{N}\left( {n + \frac{1}{2} - \frac{N}{2}} \right)\left( {k - N + \frac{1}{2}} \right)} \right\rbrack}\mspace{20mu} s_{n,k}} = {\sin\left\lbrack {\frac{\pi}{N}\left( {n + \frac{1}{2} - \frac{N}{2}} \right)\left( {k - N + \frac{1}{2}} \right)} \right\rbrack}}}}} & (3)\end{matrix}$where x_(i) (i=1, 2, . . . , 8) represents each MDCT coefficient.Further, y_(j) (i=1, 2, . . . , 8) represents the real component of eachQMF coefficient, and y_(j) (i=9, 10, . . . , 16) represents theimaginary component of each QMF coefficient. Each element C_(n,k) in thetransform matrix represents the basis function used for the IMDCT in theIMDET, and is a cosine function. On the other hand, each element S_(n,k)in the transform matrix represents the basis function used for the IMDSTin the IMDET, and is a sine function.

Since the basis functions used in the IMDET are trigonometric functions,as noted above, the basis functions have a periodicity. The periodicitywill be described with reference to FIG. 8. In FIG. 8, graph 800represents the basis function of the IMDCT for k=0. The basis functionc_(0,k) to c_(7,k) contains half of the period of the cosine function;then, noting the first half interval of the basis function, i.e., fromc_(0,k) to c_(3,k), it is seen that the interval c_(0,k) to c_(3,k) issymmetrical about the midpoint of that interval. Hence, c_(0,k)=c_(3,k)and c_(1,k)=c_(2,k) hold. For the second half interval c_(4,k) toc_(7,k) also, the basis function in absolute value is symmetrical aboutthe midpoint of that interval, though the sign is inverted. Hence,c_(4,k)=c_(7,k) and c_(5,k)=c_(6,k) hold. Such symmetry also holds whenk is not 0. Likewise, the basis function s_(0,k) to s_(7,k) of the IMDSTis a sine function, and as can be seen from the equation (3), since itcontains half of the period of the sine function, the interval s_(0,k)to s_(3,k) and the interval s_(4,k) to s_(7,k) are each symmetrical.

Accordingly, if the computation results are obtained for half of therows in the transform matrix of the equation (3), the computationresults can be used for the other half of the rows. More specifically,the inverse modified discrete exponential transform unit 22 need onlyperform computations only for the rows corresponding to either the firsthalf or second half of the first half of the sequence of the QMFcoefficients to be computed by the equation (3) and the rowscorresponding to either the first half or second half of the second halfof the sequence of the QMF coefficients. For example, y4 (the row ofn=3) and y3 (the row of n=2) can be computed using the computationresults of y1 (the row of n=0) and y2 (the row of n=1), respectively.Likewise, y8 (the row of n=7) and y7 (the row of n=6) can be computedusing the computation results of y5 (the row of n=4) and y6 (the row ofn=5), respectively.

Referring to FIG. 9, the difference between the basis function of theIMDCT and the basis function of the IMDST will be described below. InFIG. 9, graph 900 represents the basis function of the IMDCT for k=0. Onthe other hand, graph 910 represents the basis function of the IMDST fork=0. The only difference between the basis function, c_(0,k) to c_(7,k),of the IMDCT and the basis function, s_(0,k) to s_(7,k), of the IMDST isthat one is a cosine function and the other is a sine function.Therefore, noting the vertical columns in the transform matrix of theequation (3), it is seen that the basis function, c_(0,k) to c_(7,k), ofthe IMDCT and the basis function, s_(0,k) to s_(7,k), of the IMDST areidentical in shape, the only difference being that the phase is shiftedby one quarter of the period. Accordingly, the absolute values of thebasis function, s_(0,k) to s_(7,k), of the IMDST are each equal to oneof the absolute values of the basis function, c_(0,k) to c_(7,k), of theIMDCT. In other words, the inverse modified discrete exponentialtransform unit 22 need only perform computations only for the IMDCT(i.e., the upper half of the transform matrix) or the IMDST (i.e., thelower half of the transform matrix) in the equation (3); then, theresults of the computations can be used for the other half. For example,using the computation results of the IMDCT performed on the first halfof the sequence of the QMF coefficients to be computed by the equation(3), the inverse modified discrete exponential transform unit 22 cancompute the IMDST of the second half of the sequence of the QMFcoefficients. Conversely, using the computation results of the IMDCTperformed on the second half of the sequence of the QMF coefficients tobe computed by the equation (3), the inverse modified discreteexponential transform unit 22 can compute the IMDST of the first half ofthe sequence of the QMF coefficients. Similarly, using the computationresults of the IMDST performed on the first half and the second half ofthe sequence of the QMF coefficients, the inverse modified discreteexponential transform unit 22 can compute the IMDCT of the second halfand the first half of the sequence of the QMF coefficients,respectively.

Accordingly, by performing multiplications with the MDCT coefficientsonly on one quarter of the elements contained in the matrix of theequation (3), the inverse modified discrete exponential transform unit22 can accomplish the computation of the IMDET.

To compute the QMF coefficients by performing multiplications with theMDCT coefficients only on some of the elements contained in the matrixof the equation (3) as described above, the inverse modified discreteexponential transform unit 22 includes a storage unit 31, a localmultiplication unit 32, and a coefficient computing unit 33.

The storage unit 31 includes, for example, a nonvolatile read-onlymemory circuit and a readable/writable volatile memory circuit. Then,for each of the various lengths of interval to which the IMDET isapplied, the storage unit 31 stores a table representing the elements ofthe basis functions by which the MDCT coefficients are multiplied. Eachtable stores the values of the basis functions used to compute eitherthe first half or the second half of the first half of the intervalcontaining the QMF coefficients to be computed, and the values of thebasis functions used to compute either the first half or the second halfof the second half of that interval. The values of the basis functionsmay be the values of the basis functions for the IMDCT, i.e., the valuesused to compute the real components of the QMF coefficients, or may bethe values of the basis functions for the IMDST, i.e., the values usedto compute the imaginary components of the QMF coefficients. Forexample, the table for the case where the length of interval to whichthe IMDET is applied is 8, that is, N=4, as indicated by the equation(3), stores the elements of the basis functions (c_(0,k), c_(1,k),c_(4,k), and c_(5,k)) in the first, second, fifth, and sixth rows of thematrix in the equation (3). When the length of interval to which theIMDET is applied is 4, i.e., when N=2, the matrix of the basis functionshas elements arranged in eight rows and four columns. Of these elements,the elements in the upper four rows are the IMDCT basis functionsc_(n,k) (n=0, . . . , 3, k=0, . . . , 3), and the elements in the lowerfour rows are the IMDST basis functions s_(n,k) (n=0, . . . , 3, k=0, .. . , 3). The table for N=2 stores the elements of the basis functions(c_(0,k) and c_(2,k)) carried in the first and third rows.

The storage unit 31 also stores intermediate computed values from thelocal multiplication unit 32 so that the values can be used by thecoefficient computing unit 33.

The local multiplication unit 32 retrieves the desired table from amongthe tables stored in the storage unit 31 according to the frame lengthof the residual signal. The local multiplication unit 32 multiplies theelement of each basis function stored in the table by the correspondingMDCT coefficient. Then, for each row of the transform matrix on whichthe multiplications are done, the local multiplication unit 32 computesthe sum Σc_(i,2k)*x[2k+1] (k=0, 1, . . . , 2N−1) of the products of theMDCT coefficients and the elements in the odd-numbered columns and thesum Σc_(i,2k+1)*x[2k+1] of the products of the MDCT coefficients and theelements in the even-numbered columns. The local multiplication unit 32stores these sums as the intermediate computed values in the storageunit 31.

FIG. 10 is a diagram for explaining the processing that the localmultiplication unit 32 performs when the length of interval is 8 (N=4).In the matrix 1000 containing the products of the transform matrix andthe MDCT coefficient sequence, the local multiplication unit 32calculates the elements c_(i,2k)*x[2k+1] in the odd-numbered columnsenclosed by solid lines for each of the first, second, fifth, and sixthrows. Then, the local multiplication unit 32 computes the sumΣc_(i,2k)*x[2k+1] of these elements as y_(i)odd. Further, the localmultiplication unit 32 calculates the elements c_(i,2k+1)*x[2(k+1)] inthe even-numbered columns enclosed by dashed lines. Then, the localmultiplication unit 32 computes the sum Σc_(i,2k+1)*x[2(k+1)] of theseelements as y_(i)even.

The coefficient computing unit 33 computes the real and imaginarycomponents of each QMF coefficient by a butterfly operation using theintermediate computed values stored in the storage unit 31.

FIG. 11 is a diagram for explaining how the butterfly operation isperformed by the coefficient computing unit 33. In the illustratedexample, of a plurality of subintervals among which the QMF coefficientsequence is divided into four quarters so that the values of the basisfunctions used to compute each QMF coefficient are symmetrically placed,the intermediate computed values for the first and third subintervalsare already obtained from the local multiplication unit 32. In otherwords, of the eight blocks into which the matrix 1100 containing theproducts of the IMDET transform matrix and the MDCT coefficients hasbeen equally divided across the vertical direction, the sum y_(i)odd ofthe elements in the odd-numbered columns and the sum y_(i)even of theelements in the even-numbered columns are already computed for each rowin the first and third blocks 1101 and 1103. The numbers indicated atthe right edge of each block indicate the row numbers for which theintermediate computed values to be used for the computation of thecorresponding rows have been calculated by the local multiplication unit32.

For each row in the blocks 1101 and 1103, the coefficient computing unit33 can compute the real component of the corresponding QMF coefficientby simply adding the sum y_(i)odd of the elements in the odd-numberedcolumns to the sum y_(i)even of the elements in the even-numberedcolumns.

On the other hand, the value of the QMF coefficient corresponding toeach row in the second block 1102 is symmetrical in position to thevalue of the QMF coefficient corresponding to each row in the firstblock 1101. For example, when N=4, each block contains two rows. As aresult, the real component y3 of the QMF coefficient corresponding tothe upper row in the block 1102, i.e., the third row in the matrix 1100,is equal to the real component y2 of the QMF coefficient correspondingto the lower row in the block 1101, i.e., the second row in the matrix1100. Likewise, the real component y4 of the QMF coefficientcorresponding to the lower row in the block 1102, i.e., the fourth rowin the matrix 1100, is equal to the real component y1 of the QMFcoefficient corresponding to the upper row in the block 1101, i.e., thefirst row in the matrix 1100. Accordingly, the coefficient computingunit 33 substitutes the real component of the QMF coefficientcorresponding to each row in the block 1101 for the real component ofthe QMF coefficient in the corresponding row in the block 1102. Forexample, when N=4, y3=y2 and y4=y1.

Similarly, the value of the QMF coefficient corresponding to each row inthe fourth block 1104 is symmetrical in position to the value of the QMFcoefficient corresponding to each row in the third block 1103, but thesign of the value is inverted. Accordingly, the coefficient computingunit 33 inverts the sign of the real component of the QMF coefficientcorresponding to each row in the block 1103, and substitutes thesign-inverted real component for the real component of the QMFcoefficient in the corresponding row in the block 1104. For example,when N=4, y8=−y5 and y7=−y6.

Further, since the IMDCT basis function and the IMDST basis function areshifted in phase by one-quarter of the period relative to each other,the coefficient computing unit 33 can compute the value of each row inthe eighth block 1108 by using the intermediate computed values of thecorresponding row in the first block 1101. When N=4, the imaginarycomponent y16 of the QMF coefficient corresponding to the lower row inthe eighth block 1108, i.e., the 16th row in the matrix 1100, is equalto the value (y₁even−y₁odd) obtained by subtracting the sum y₁odd of theelements in the odd-numbered columns in the upper row y1 in the block1101 from the sum y₁even of the elements in the even-numbered columns.Similarly, the imaginary component y15 of the QMF coefficientcorresponding to the 15th row in the matrix 1100 is equal to the value(y₂even−y₂odd) obtained by subtracting the sum y₂odd of the elements inthe odd-numbered columns in the lower row y2 in the block 1101 from thesum y₂even of the elements in the even-numbered columns. Further, thecoefficient computing unit 33 can compute the imaginary component of theQMF coefficient corresponding to each row in the sixth block 1106 byusing the intermediate computed values of the corresponding row in thethird block 1103. More specifically, the imaginary component y12 of theQMF coefficient corresponding to the 12th row in the matrix 1100 isequal to the value (y₅even−y₅odd) obtained by subtracting the sum y₅oddof the elements in the odd-numbered columns in the upper row y5 in theblock 1103 from the sum y₅even of the elements in the even-numberedcolumns. Similarly, the imaginary component y11 of the QMF coefficientcorresponding to the 11th row in the matrix 1100 is equal to the value(y₆even−y₆odd) obtained by subtracting the sum y₆odd of the elements inthe odd-numbered columns in the lower row y6 in the block 1103 from thesum y₆even of the elements in the even-numbered columns.

Further, the value of the QMF coefficient corresponding to each row inthe seventh block 1107 is symmetrical in position to the value of theQMF coefficient corresponding to each row in the eighth block 1108. Forexample, when N=4, the imaginary component y13 of the QMF coefficientcorresponding to the upper row in the block 1107, i.e., the 13th row inthe matrix 1100, is equal to the imaginary component y16 of the QMFcoefficient corresponding to the lower row in the block 1108, i.e., the16th row in the matrix 1100. Likewise, the imaginary component y14 ofthe QMF coefficient corresponding to the lower row in the block 1107,i.e., the 14th row in the matrix 1100, is equal to the imaginarycomponent y15 of the QMF coefficient corresponding to the 15th row inthe matrix 1100.

Similarly, the value of the QMF coefficient corresponding to each row inthe fifth block 1105 is symmetrical in position to the value of the QMFcoefficient corresponding to each row in the sixth block 1106, but thesign of the value is inverted. Accordingly, the coefficient computingunit 33 inverts the sign of the imaginary component of the QMFcoefficient corresponding to each row in the block 1106, and substitutesthe thus sign-inverted imaginary component for the imaginary componentof the QMF coefficient in the corresponding row in the block 1105. Forexample, when N=4, y9=−y12 and y10=−y11.

FIG. 12 is a diagram for explaining the butterfly operation according toa modified example when intermediate computed values are calculated bythe local multiplication unit 32 for the fifth and seventh blocks amongthe eight blocks into which the matrix 1200 containing the products ofthe IMDET transform matrix and the MDCT coefficients has been equallydivided across the vertical direction.

In this case, since the fifth block 1205 and the seventh block 1207 eachcorrespond to IMDST, the basis functions are sine functions.Accordingly, the sum Σs_(i,2k)*x[2k+1] of the elements in theodd-numbered columns is computed as y_(i)odd by the local multiplicationunit 32 for each of the 9th, 10th, 13th, and 14th rows in the matrix ofthe products of the transform matrix and the MDCT coefficients.Similarly, the sum Σs_(i,2k+1)*x[2(k+1)] of the elements in theeven-numbered columns is computed as y_(i)even by the localmultiplication unit 32. For each row in the blocks 1205 and 1207, thecoefficient computing unit 33 can compute the imaginary component yi(i=9, 10, 13, 14) of the corresponding QMF coefficient by simply addingthe sum y_(i)odd of the elements in the odd-numbered columns to the sumy_(i)even of the elements in the even-numbered columns.

On the other hand, the value of the QMF coefficient corresponding toeach row in the eighth block 1208 is symmetrical in position to thevalue of the QMF coefficient corresponding to each row in the seventhblock 1207. Accordingly, when N=4, for example, the coefficientcomputing unit 33 computes the imaginary components y15 and y16 of theQMF coefficients corresponding to the respective rows in the block 1208as y15=y14 and y16=y13, respectively.

Similarly, the value of the QMF coefficient corresponding to each row inthe sixth block 1206 is symmetrical in position to the value of the QMFcoefficient corresponding to each row in the fifth block 1205, but thesign of the value is inverted. Accordingly, when N=4, for example, thecoefficient computing unit 33 computes the imaginary components y11 andy12 of the QMF coefficients corresponding to the respective rows in theblock 1206 as y11=−y10 and y12=−y9, respectively.

Further, since the IMDCT basis function and the IMDST basis function areshifted in phase by one-quarter of the period relative to each other,the coefficient computing unit 33 can compute the value of each row inthe second block 1202 by using the intermediate computed values of thecorresponding row in the seventh block 1207. When N=4, the realcomponent y4 of the QMF coefficient corresponding to the lower row inthe second block 1202, i.e., the fourth row in the matrix 1200, is equalto the value (y₁₃odd−y₁₃even) obtained by subtracting the sum y₁₃even ofthe elements in the even-numbered columns in the upper row y13 in theblock 1207 from the sum y₁₃odd of the elements in the odd-numberedcolumns. Similarly, the real component y3 of the QMF coefficientcorresponding to the third row in the matrix 1200 is equal to the value(y₁₄odd−y₁₄even) obtained by subtracting the sum y₁₄even of the elementsin the even-numbered columns in the lower row y14 in the block 1207 fromthe sum y₁₄odd of the elements in the odd-numbered columns. Further, thecoefficient computing unit 33 can compute the real component of the QMFcoefficient corresponding to each row in the fourth block 1204 by usingthe intermediate computed values of the corresponding row in the fifthblock 1205. More specifically, the real component y7 of the QMFcoefficient corresponding to the seventh row in the matrix 1200 is equalto the value (y₁₀odd−y₁₀even) obtained by subtracting the sum y₁₀even ofthe elements in the even-numbered columns in the lower row y10 in theblock 1205 from the sum y₁₀odd of the elements in the odd-numberedcolumns. Similarly, the real component y8 of the QMF coefficientcorresponding to the eighth row in the matrix 1200 is equal to the value(y₉odd−y₉even) obtained by subtracting the sum y₉even of the elements inthe even-numbered columns in the upper row y9 in the block 1205 from thesum y₉odd of the elements in the odd-numbered columns.

Further, the value of the QMF coefficient corresponding to each row inthe first block 1201 is symmetrical in position to the value of the QMFcoefficient corresponding to each row in the second block 1202.Accordingly, when N=4, for example, the coefficient computing unit 33computes the real components y1 and y2 of the QMF coefficientscorresponding to the respective rows in the block 1201 as y1=y4 andy2=y3, respectively.

Similarly, the value of the QMF coefficient corresponding to each row inthe third block 1203 is symmetrical in position to the value of the QMFcoefficient corresponding to each row in the fourth block 1204, but thesign of the value is inverted. Accordingly, when N=4, for example, thecoefficient computing unit 33 computes the real components y5 and y6 ofthe QMF coefficients corresponding to the respective rows in the block1203 as y5=−y8 and y6=−y7, respectively.

As described above, to compute the real components in the first halfinterval of the QMF coefficient sequence and the imaginary components inthe second half, the products of the basis functions and thecorresponding MDF coefficients need be calculated only for the firsthalf or second half of either one of the first and second halfintervals. Similarly, to compute the imaginary components in the firsthalf interval of the sequence of the QMF coefficients and the realcomponents in the second half, the products of the basis functions andthe corresponding MDF coefficients need be calculated only for the firsthalf or second half of either one of the first and second halfintervals.

The coefficient computing unit 33 passes the real and imaginarycomponents of the QMF coefficients to the coefficient adjusting unit 23.

The coefficient adjusting unit 23 obtains the QMF coefficients of theresidual signal by combining the real components and imaginarycomponents of the QMF coefficients output from the inverse modifieddiscrete exponential transform unit 22. More specifically, thecoefficient adjusting unit 23 computes each QMF coefficient inaccordance with the following equation.

$\begin{matrix}{\mspace{79mu}{{{z\left\lbrack {n,f} \right\rbrack} = \left\{ {{X\left\lbrack {n,f} \right\rbrack} + {j\;{Y\left\lbrack {n,f} \right\rbrack}}} \right\}}{Z = \left\lbrack {{\exp\left( {{- \frac{j\;\pi}{256}}\left( {{258 \cdot 0} + 385} \right)} \right)}{z\left\lbrack {n,1} \right\rbrack}{\exp\left( {{- \frac{j\;\pi}{256}}\left( {{258 \cdot 1} + 385} \right)} \right)}{z\left\lbrack {n,2} \right\rbrack}\mspace{14mu}\ldots\mspace{14mu}{\exp\left( {{- \frac{j\;\pi}{256}}\left( {{258 \cdot \left( {f - 1} \right)} + 385} \right)} \right)}{z\left\lbrack {n,f} \right\rbrack}} \right\rbrack}}} & (4)\end{matrix}$where X[n,f] is the real component of the QMF coefficient obtained bythe butterfly IMDCT of the MDCT coefficient, and Y[n,f] is the imaginarycomponent of the QMF coefficient obtained by the butterfly IMDST of theMDCT coefficient. Z[n,f] is the resulting QMF coefficient. f denotes thefrequency band to which the butterfly IMDCT and butterfly IMDST areapplied.

FIG. 13 is an operation flowchart illustrating the orthogonal transformprocess performed by the orthogonal transform unit 16. The orthogonaltransform unit 16 performs the orthogonal transform process inaccordance with the following flowchart for each computation intervalcorresponding to one frequency band.

The windowing unit 21 in the orthogonal transform unit 16 multiplies theMDCT coefficients of the residual signal by the windowing function andthe gain (step S101). Then, the windowing unit 21 passes the MDCTcoefficients, each multiplied by the windowing function and the gain, tothe local multiplication unit 32 in the inverse modified discreteexponential transform unit 22 contained in the orthogonal transform unit16.

The local multiplication unit 32 accesses the storage unit 31 toretrieve, according to the length of the computation interval, a tablethat stores the values of the basis functions contained in a designatedone of the subintervals among which the QMF coefficient sequence isdivided so that the values of the basis functions used to compute theQMF coefficients are symmetrically placed (step S102). Then, the localmultiplication unit 32 computes the sum of the products of the basisfunctions in the odd-numbered columns corresponding to the realcomponent of the QMF coefficient contained in the designated subintervaland the corresponding MDCT coefficients and the sum of the products ofthe basis functions in the even-numbered columns and the correspondingMDCT coefficients. The local multiplication unit 32 stores the resultsof the computations as the intermediate computed values in the storageunit 31 (step S103).

For the designated subinterval, the coefficient computing unit 33computes the real component of the QMF coefficient contained in thatsubinterval by summing the sum of the products of the basis functions inthe odd-numbered columns and the corresponding MDCT coefficients withthe sum of the products of the basis functions in the even-numberedcolumns and the corresponding MDCT coefficients (step S104). Further,for a subinterval other than the designated subinterval, the coefficientcomputing unit 33 computes the real component of the QMF coefficient bya butterfly operation using the intermediate computed values stored inthe storage unit 31 and utilizing the symmetry of the basis functions(step S105). Further, from the intermediate computed values of thedesignated subinterval, the coefficient computing unit 33 compensatesfor the phase difference between the IMDCT basis function and the IMDSTbasis function, and computes the imaginary component of the QMFcoefficient contained in each subinterval by utilizing the symmetry ofthe basis functions between the subintervals (step S106).

The coefficient adjusting unit 23 in the orthogonal transform unit 16obtains each QMF coefficient by combining the real and imaginarycomponents of the QMF coefficient (step S107). Then, the orthogonaltransform unit 16 terminates the orthogonal transform process. In theabove step S103, the local multiplication unit 32 may compute the sum ofthe products of the basis functions in the odd-numbered columnscorresponding to the imaginary component of the QMF coefficientcontained in the designated subinterval and the corresponding MDCTcoefficients and the sum of the products of the basis functions in theeven-numbered columns and the corresponding MDCT coefficients, and maytake them as the intermediate computed values. In this case, thecoefficient computing unit 33 computes the imaginary component of theQMF coefficient contained in each subinterval in the above steps S104and S105, and computes the real component of the QMF coefficientcontained in each subinterval in the above step S106.

FIG. 14 is an operation flowchart of the audio decoding processperformed by the audio decoding apparatus 1. The audio decodingapparatus 1 decodes the audio signal on a frame-by-frame basis inaccordance with the following flowchart.

The demultiplexing unit 11 demultiplexes the main signal code such asthe AAC code and SBR code, the spatial information code, and theresidual signal code from the encoded data stream (step S201). The mainsignal decoding unit 12 reconstructs the stereo signal by decoding themain signal code received from the demultiplexing unit 11 (step S202).The time-frequency transform unit 13 transforms the stereo signal intoQMF coefficients in the time-frequency domain by applying a QMF filterbank (step S203).

On the other hand, the spatial information decoding unit 14 reconstructsthe spatial information by decoding the spatial information codereceived from the demultiplexing unit 11 (step S204). The spatialinformation decoding unit 14 passes the reconstructed spatialinformation to the upmixing unit 17.

The residual signal decoding unit 15 reconstructs the MDCT coefficientsof the residual signal by decoding the residual signal code receivedfrom the demultiplexing unit 11 (step S205). The orthogonal transformunit 16 computes the QMF coefficients of the residual signal by applyingthe butterfly IMDET to the MDCT coefficients of the residual signal byusing the intermediate computed values of some of the QMF coefficientsfor computation of other QMF coefficients by utilizing the symmetry ofthe basis functions (step S206).

The upmixing unit 17 reconstructs the QMF coefficients for each channelof the original audio signal by upmixing the QMF coefficients of thestereo signal and the QMF coefficients of the residual signal by usingthe spatial information (step S207). The frequency-time transform unit18 reconstructs each channel of the audio signal by frequency-timetransforming the QMF coefficients of the corresponding channel (stepS208).

Then, the audio decoding apparatus terminates the audio decodingprocess.

As has been described above, the orthogonal transform apparatusaccording to the present embodiment can reduce the amount of computationof the butterfly IMDET used to transform the MDCT coefficients into theQMF coefficients, by a factor of 4 by exploiting the symmetry of thebasis functions. As a result, the audio decoding apparatus incorporatingthe above orthogonal transform apparatus can reduce the amount ofcomputation needed to transform the MDCT coefficients of the residualsignal into the QMF coefficients.

Next, an orthogonal transform apparatus according to a second embodimentwill be described.

When the IMDET is performed without applying any speed enhancingtechniques, the amount of computation of the IMDET is of the order ofthe square of the number of MDCT coefficients contained in thecomputation interval. As a result, in the foregoing embodiment, whilethe amount of computation of the IMDET can be reduced by a factor of 4,the amount of computation itself is of the order of the square of thenumber of MDCT coefficients contained in the computation interval.

On the other hand, methods are known that perform IMDCT and IMDST byusing fast Fourier transform (FFT). One such method is disclosed, forexample, in “Regular FFT-Related Transform Kernels for DCT/DST-BasedPolyphase Filter Banks” by Rolf Gluth, IEEE Acoustics, Speech, andSignal Processing, ICASSP-91, 1991, vol. 3, pp. 2205-2208. According tothe method disclosed in the above literature, IMDCT and IMDST can beimplemented by performing FFT in both the pre-processing andpost-processing that involve rotating the input signal sequence in thecomplex plane. The amount of FFT computation is of the order of N log Nwhen the number, N, of input signal points is a power of 2. Accordingly,when the number of MDCT coefficients contained in the IMDET computationinterval is a power of 2, the effect of reducing the amount ofcomputation by performing the IMDET using FFT is greater as the numberof MDCT coefficients is larger.

In view of the above, in the orthogonal transform apparatus according tothe second embodiment, switching between the IMDET method according tothe foregoing embodiment and the IMDET method that uses FFT is madebased on the length of the computation interval to which the IMDET isapplied.

FIG. 15 is a block diagram of the orthogonal transform apparatus 16′according to the second embodiment. The orthogonal transform apparatus16′ includes a windowing unit 21, a switching unit 24, an inversemodified discrete exponential transform unit 22, a second inversemodified discrete exponential transform unit 25, and a coefficientadjusting unit 23. In FIG. 15, the component elements of the orthogonaltransform apparatus 16′ are designated by the same reference numerals asthose used to designate the corresponding component elements of theorthogonal transform apparatus 16 according to the first embodimentillustrated in FIG. 7. The orthogonal transform apparatus 16′ accordingto the second embodiment differs from the orthogonal transform apparatus16 according to the first embodiment by the inclusion of the switchingunit 24 and the second inverse modified discrete exponential transformunit 25. The following description therefore deals with the switchingunit 24 and the second inverse modified discrete exponential transformunit 25.

Based on the length of the computation interval to which the IMDET isapplied, the switching unit 24 selects either the inverse modifieddiscrete exponential transform unit 22 which performs the IMDET byexploiting the symmetry of the basis functions or the second inversemodified discrete exponential transform unit 25 which performs the IMDETby using FFT.

FIG. 16 is an operation flowchart of the switching process performed bythe switching unit 24. The switching unit 24 determines whether or notthe number, M, of MDCT coefficients contained in the IMDET computationinterval is equal to or larger than 8 (step S301). If the number, M, ofMDCT coefficients is equal to or larger than 8 (Yes in step S301), theswitching unit 24 then determines whether the number, M, of MDCTcoefficients is a power 2 or not (step S302). If the number, M, of MDCTcoefficients is a power 2 (Yes in step S302), the switching unit 24directs the MDCT coefficients to the second inverse modified discreteexponential transform unit 25 that uses FFT (step S303).

On the other hand, if the number, M, of MDCT coefficients is not a power2 (No in step S302), or if the number M is smaller than 8 (No in stepS301), the switching unit 24 directs the MDCT coefficients to theinverse modified discrete exponential transform unit 22 that exploitsthe symmetry of the basis functions (step S304). After step S303 orS304, the switching unit 24 terminates the switching process.

The second inverse modified discrete exponential transform unit 25performs the IMDET of the input MDCT coefficients by using FFT.

FIG. 17 is a block diagram of the second inverse modified discreteexponential transform unit 25. The second inverse modified discreteexponential transform unit 25 includes an interchanging unit 41, aninverting unit 42, a butterfly inverse cosine transform unit 43, and abutterfly inverse sine transform unit 44. In the present embodiment, toreduce the amount of computation, the butterfly inverse cosine transformunit 43 and the butterfly inverse sine transform unit 44 employ themethod of performing the IMDCT and IMDST by using FFT.

There are differences such as described below between the butterflyIMDCT and butterfly IMDST and the conventional IMDCT and conventionalIMDST.

Generally, the butterfly IMDCT is expressed by the following equation.

$\begin{matrix}{{{ButterflyIMDCT}\lbrack n\rbrack} = {{\sqrt{\frac{1}{2\; N}}{w\lbrack n\rbrack}{\sum\limits_{k = 0}^{{2\; N} - 1}\;{{{x\lbrack k\rbrack} \cdot {\cos\left( {\frac{\pi}{N}\left( {n + n_{0}} \right)\left( {k - N + \frac{1}{2}} \right)} \right)}}0}}} \leq n < {2\; N}}} & (5)\end{matrix}$On the other hand, the conventional IMDCT is expressed by the followingequation.

$\begin{matrix}{{{IMDCT}\lbrack n\rbrack} = {{\sqrt{\frac{2}{N}}{w\lbrack n\rbrack}{\sum\limits_{k = 0}^{N - 1}\;{{{x\lbrack k\rbrack} \cdot {\cos\left( {\frac{\pi}{N}\left( {n + n_{0}} \right)\left( {k + \frac{1}{2}} \right)} \right)}}0}}} \leq n < {2\; N}}} & (6)\end{matrix}$where x[k] (k=0, 1, 2, . . . , 2N−1) are MDCT coefficients. As isapparent from the above equations (5) and (6), in the butterfly IMDCT,the number of MDCT coefficients per computation interval is twice thatin the conventional IMDCT. Further, the cosine basis functions differ inphase by (n+n₀)π. Similarly, the number of MDCT coefficients percomputation interval and the phase of the sine basis functions differbetween the butterfly IMDST and the conventional IMDST. As a result, ifthe method that uses FFT in the conventional IMDCT and IMDST weredirectly applied to the butterfly IMDCT and IMDST, the reconstructedoriginal signal (in the present embodiment, the residual signal) wouldcontain artifact signal components, resulting in a degradation of theoriginal signal.

In view of this, before performing the IMDCT and IMDST, the secondinverse modified discrete exponential transform unit 25 reorders theMDCT coefficients and invert their signs so that the number of MDCTcoefficients per computation interval and the phase of the basisfunction match the number of coefficients and the phase of the basisfunction in the conventional IMDCT or IMDST.

Referring to FIG. 18, a description will be given of the relationshipbetween the cosine basis function of the butterfly IMDCT and the cosinebasis function of the conventional IMDCT. In FIG. 18, the abscissarepresents the frequency k of the MDCT coefficient. Graph 1801represents the cosine basis function c1[k] of the conventional IMDCT,while graph 1802 represents the cosine basis function c2[k] of thebutterfly IMDCT. Since the functions c1[k] and c2 [k] respectivelycorrespond to the trigonometric function terms in the equations (5) and(6), these functions are respectively expressed by the followingequations.

$\begin{matrix}{{c\;{1\lbrack k\rbrack}} = {\cos\left( {\frac{\pi}{N}\left( {n + n_{0}} \right)\left( {k + \frac{1}{2}} \right)} \right)}} & (7) \\{{c\;{2\lbrack k\rbrack}} = {\cos\left( {\frac{\pi}{N}\left( {n + n_{0}} \right)\left( {k - N + \frac{1}{2}} \right)} \right)}} & (8)\end{matrix}$

It can be seen from FIG. 18 and the above equations (7) and (8) that,between the functions c1[k] and c2 [k], the value of k is shifted inphase by N. This means that the value of the cosine basis function c2[k]of the butterfly IMDCT on the interval [0, N−1] is equal to the value ofthe cosine basis function c1[k] of the conventional IMDCT on theinterval [N, 2N−1].

Further, the cosine basis functions c1[k] and c2[k] are equal inabsolute value but opposite in sign to the values c1[k−2N] and c2[k−2N]of the respective functions when the value of k differs by 2N. In otherwords, the following relation holds between the cosine basis functionsc1[k] and c2[k].c1[k]=c2[k+N] 0≦k<Nc1[k]=−c2[k−N] N≦k<2N  (9)

Hence, the following equation holds.

$\begin{matrix}\begin{matrix}{{{ButterflyIMDCT}^{\prime}\lbrack n\rbrack} = {{\sum\limits_{k = 0}^{{2\; N} - 1}\;{{{x\lbrack k\rbrack} \cdot c}\;{2\lbrack k\rbrack}\; 0}} \leq n < {2\; N}}} \\{= {{\sum\limits_{k = 0}^{N - 1}\;{{{x\lbrack k\rbrack} \cdot c}\;{2\lbrack k\rbrack}}} + {\sum\limits_{k = N}^{{2\; N} - 1}\;{{{x\lbrack k\rbrack} \cdot c}\;{2\lbrack k\rbrack}}}}} \\{= {{\sum\limits_{k = 0}^{N - 1}\;{{{x\left\lbrack {k + N} \right\rbrack} \cdot c}\;{1\lbrack k\rbrack}}} + {\sum\limits_{k = N}^{{2\; N} - 1}\;{{{- {x\left\lbrack {k - N} \right\rbrack}} \cdot c}\;{1\lbrack k\rbrack}}}}}\end{matrix} & (10)\end{matrix}$

As can be seen from the equation (10), if the MDCT coefficients x[k](k=0, 1, . . . , N−1) in the first half of the interval are interchangedwith the MDCT coefficients x[k] (k=N, N+1, . . . , 2N−1) in the secondhalf, it becomes possible to apply the cosine basis function c1[k] ofthe conventional IMDCT to the first half after the interchange. On theother hand, for the MDCT coefficients contained in the second half afterthe interchange, if their signs are inverted, it becomes possible toapply the cosine basis function c1[k] of the conventional IMDCT. Thefirst and second halves of the interval are each equal in length to theinterval to which the conventional IMDCT is applied. Accordingly, theconventional IMDCT can be applied to each of the first and secondhalves.

A similar relation holds between the sine basis function of thebutterfly IMDST and the sine basis function of the conventional IMDST.Accordingly, for the butterfly IMDST also, if the MDCT coefficients inthe first half of the computation interval are interchanged with theMDCT coefficients in the second half, and if the sign of the MDCTcoefficients contained in the first half after the interchange isinverted, it becomes possible to apply the conventional IMDCT to each ofthe first and second halves.

Therefore, the interchanging unit 41 interchanges the MDCT coefficientsin the first half of the computation interval with the MDCT coefficientsin the second half. The processing performed by the interchanging unit41 will be described with reference to FIG. 19. The interchanging unit41 reorders the MDCT coefficients x[k], each multiplied by the windowingfunction and the gain, by interchanging the order of the first half andthe second half, as indicated by arrows in FIG. 19, and thereby obtainsthe MDCT coefficients x′[k] after the interchange. The process ofinterchanging is expressed by the following equation.x′[k]=x[k−N] N≦k<2Nx′[k]=x[k+N] 0≦k<N  (11)

The interchanging unit 41 passes the MDCT coefficients x′[k] containedin the first half after the interchange, i.e., the MDCT coefficientsinitially contained in the second half, to an inverse cosine transformunit 51-1 in the butterfly inverse cosine transform unit 43 and aninverse sine transform unit 53-1 in the butterfly inverse sine transformunit 44. At the same time, the interchanging unit 41 passes the MDCTcoefficients x′[k] contained in the second half after the interchange,i.e., the MDCT coefficients initially contained in the first half, tothe inverting unit 42.

The inverting unit 42 inverts the sign of the MDCT coefficients x′[k]contained in the second half after the interchange. The inverting unit42 passes the MDCT coefficients x′[k] inverted in sign to an inversecosine transform unit 51-2 in the butterfly inverse cosine transformunit 43 and an inverse sine transform unit 53-2 in the butterfly inversesine transform unit 44.

The butterfly inverse cosine transform unit 43 computes the realcomponents of the QMF coefficients by performing the conventional IMDCTusing FFT after performing processing such as reordering the MDCTcoefficients within the computation interval, rather than directlyimplementing the butterfly IMDCT. Referring back to FIG. 17, thebutterfly inverse cosine transform unit 43 includes an adder 52 inaddition to the inverse cosine transform units 51-1 and 51-2.

Similarly, the butterfly inverse sine transform unit 44 computes theimaginary components of the QMF coefficients by performing theconventional IMDST using FFT after performing processing such asreordering the MDCT coefficients within the computation interval. Forthis purpose, the butterfly inverse sine transform unit 44 includes anadder 54 in addition to the inverse sine transform units 53-1 and 53-2.

The following description deals only with the butterfly inverse cosinetransform unit 43. By simply changing the basis functions used for thetransform from the cosine functions to the sine functions, the butterflyinverse sine transform unit 44 can accomplish the butterfly IMDST byapplying the conventional IMDST using FFT to the MDCT coefficients in amanner similar to the butterfly inverse cosine transform unit 43.

The inverse cosine transform unit 51-1 performs the IMDCT correspondingto the first term on the right-hand side of the equation (10) by usingFFT. On the other hand, the inverse cosine transform unit 51-2 performsthe IMDCT corresponding to the second term on the right-hand side of theequation (10) by using FFT. The inverse cosine transform units 51-1 and51-2 employ, for example, the technique disclosed in the earlier cited“Regular FFT-Related Transform Kernels for DCT/DST-Based PolyphaseFilter Banks” by Rolf Gluth, IEEE Acoustics, Speech, and SignalProcessing, ICASSP-91, 1991, vol. 3, pp. 2205-2208. Since the onlydifference between the inverse cosine transform units 51-1 and 51-2 isthe data to be processed, the following description deals only with theinverse cosine transform unit 51-1.

FIG. 20 is a block diagram of the inverse cosine transform unit 51-1. Toimplement the technique disclosed in the above literature “RegularFFT-Related Transform Kernels for DCT/DST-Based Polyphase Filter Banks”by Rolf Gluth, IEEE Acoustics, Speech, and Signal Processing, ICASSP-91,1991, vol. 3, pp. 2205-2208, the inverse cosine transform unit 51-1includes a pre-rotation unit 61, a fast Fourier transform unit 62, and apost-rotation unit 63.

To narrow the range of computation by exploiting the symmetry of thetrigonometric basis functions, the pre-rotation unit 61 obtains acomposite function f[k] by compositing the input MDCT coefficients x′[k]in four groups in accordance with the following equation.f[k]=(x[2k]+x[2N−2k−1])−j(x[N+2k]+x[N−2k−1]) (0≦k<N/2)  (12)Then, the pre-rotation unit 61 rotates the composite function f[k] inthe complex plane by one-eighth of a revolution in accordance with thefollowing equation.

$\begin{matrix}{{{f^{\prime}\lbrack k\rbrack} = {\beta_{twiddle} \cdot {f\lbrack k\rbrack}}}{\beta_{twiddle} = {\exp\left( {{- j}\frac{\pi}{N}\left( {k + \frac{1}{8}} \right)} \right)}}} & (13)\end{matrix}$

The pre-rotation unit 61 passes the rotated composite function f′[k] tothe fast Fourier transform unit 62.

The fast Fourier transform unit 62 performs the FFT of the compositefunction f′[k]. The fast Fourier transform unit 62 can apply any ofvarious computational methods known as FFT. The fast Fourier transformunit 62 passes the coefficients F[n] obtained by the FFT to thepost-rotation unit 63.

The post-rotation unit 63 computes coefficients F[n] in accordance withthe following equation by rotating the coefficients F[n] by one-eighthof a revolution in the direction opposite to the direction of therotation applied by the pre-rotation unit 61.

$\begin{matrix}{{{F^{\prime}\lbrack n\rbrack} = {{- \beta_{twiddle}} \cdot {F\lbrack n\rbrack}}}{\beta_{twiddle} = {\exp\left( {{- j}\frac{\pi}{N}\left( {k + \frac{1}{8}} \right)} \right)}}} & (14)\end{matrix}$

The post-rotation unit 63 transforms the coefficients F′[n] in thecomplex plane into the coefficients F″[n] in the real plane inaccordance with the following equation.F′[2n]=Re(F′[n])F″[N−1−2n]=−Im(F′[n])F″[N+2n]=Im(F′[n])F″[2N−1−2n]=−Re(F′[n]) (0n≦N/2)  (15)where the function Re(x) is a function that outputs the real componentof the variable x, and the function Im(x) is a function that outputs theimaginary component of the variable x. By multiplying the coefficientsF″[n] by a windowing function for the conventional IMDCT, for example, aKaiser-Bessel window, and a gain (1/N)^(1/2), the post-rotation unit 63obtains coefficients equivalent to the coefficients obtained by applyingthe IMDCT to the MDCT coefficients x′[k].

The adder 52 adds the coefficients output from the inverse cosinetransform unit 51-1 to the corresponding coefficients output from theinverse cosine transform unit 51-2. This completes the calculation ofthe right-hand side of the equation (10), completing the butterfly IMDCTof the MDCT coefficients, and the real components of the QMFcoefficients are thus obtained. The adder 52 passes the real componentsof the QMF coefficients to the coefficient adjusting unit 23.

The butterfly inverse sine transform unit 44 computes the imaginarycomponents of the QMF coefficients by performing the IMDST using FFT ina manner similar to the butterfly inverse cosine transform unit 43. Thebutterfly inverse sine transform unit 44 passes the imaginary componentsof the QMF coefficients to the coefficient adjusting unit 23.

When the real and imaginary components of the QMF coefficients arereceived from the second inverse modified discrete exponential transformunit 25, the coefficient adjusting unit 23 can likewise compute the QMFcoefficients by combining the real and imaginary components inaccordance with the earlier given equation (4).

The following table is a table that indicates the amount of computationper IMDET according to the present embodiment when the number, M, ofMDCT coefficients contained in one computation interval is (2N).

TABLE 1 AMOUNT OF COMPUTATION PER IMDET NUMBER OF MULTIPLICATIONS NUMBEROF ADDITIONS LENGTH OF M(= 2N) M M M INTERVAL (N IS EVEN) 8 4 (N IS ODD)6 (N IS EVEN) 8 4 (N IS ODD) 6 USING FFT${{Mlog}\frac{M}{4}} + \frac{M}{2} + 24$  36 26 — —${\frac{3M}{2}\log\frac{M}{4}} + \frac{3M}{2} + 8$  32 14 — —CONVENTIONAL 2M² 128 32 2M² 72 2M² − 2M 112 24 2M² − 2M 60 (DEFININGEQUATION) UTILIZING SYMMETRY $\frac{M^{2}}{2}$  32  8$\frac{M^{2}}{2} - M$ 12 $\frac{M^{2}}{2}$  32  8$\frac{M^{2}}{2} + M - 2$ 22

As seen in the table, the amount of computation per butterfly IMDCTaccording to the second inverse modified discrete exponential transformunit 25 that uses FFT is of the order of M log M which is equivalent tothe amount of computation of the FFT. On the other hand, the amount ofcomputation per IMDET according to the inverse modified discreteexponential transform unit 22 that utilizes the symmetry of the basisfunctions is of the order of M².

FIG. 21 is a diagram of graphs illustrating the relationship between thenumber, M, of MDCT coefficients contained in the computation intervaland the amount of computation for the IMDET that utilizes the symmetryof the basis functions in comparison with that for the IMDET that usesFFT. Graph 2100 represents the relationship between the number, M, ofMDCT coefficients and the number of multiplications for the IMDET thatutilizes the symmetry of the basis functions, and graph 2110 representsthe relationship between the number, M, of MDCT coefficients and thenumber of multiplications for the IMDET that uses FFT. As is apparentfrom FIG. 21, if M is smaller than 8, the amount of computation issmaller when the IMDET is performed by utilizing the symmetry of thebasis functions than when it is performed by using FFT. This is becausethe computational burden associated with the pre-processing andpost-processing becomes relatively large in the method that performs theIMDET by using FFT. Accordingly, in the present embodiment, theswitching unit 24 selects the inverse modified discrete exponentialtransform unit 22 so that the IMDET is performed by utilizing thesymmetry of the basis functions when M is smaller than 8 or when M isnot a power of 2. Especially, in the AAC coding scheme or the likeapplied to the residual signal, since a short frame is used in the caseof an attack sound, the number of MDCT coefficients contained in theIMDET computation interval may become smaller than 8. Therefore, in theorthogonal transform apparatus according to the present embodiment andthe audio decoding apparatus incorporating the orthogonal transformapparatus, the amount of computation can be reduced compared with thecase where the IMDET is performed using FFT when short frames are usedto encode the residual signal. On the other hand, in such cases as whenthe residual signal is AAC encoded using relatively long frames, theorthogonal transform apparatus and the audio decoding apparatus canreduce the amount of computation of the IMDET by performing the IMDETusing FFT.

In a modified example, the local multiplication unit 32 may calculatethe intermediate computed values by only calculating the productsbetween the basis function values corresponding to the real or imaginarycomponent of the QMF coefficient contained in the designated subintervalof the QMF coefficient sequence and the corresponding MDF coefficients.In this case, the coefficient computing unit 33 need only compute, foreach QMF coefficient, the sum of the products of the MDF coefficientsand the basis function values in the odd-numbered columns and the sum ofthe products of the MDF coefficients and the basis function values inthe even-numbered columns. In this modified example also, since thenumber of computations involved in computing the products of the basisfunction values and the MDF coefficients can be reduced to one quarterof the number of computations involved in computing the products of thebasis function values and the MDF coefficients in the conventionalIMDET, the amount of computation of the entire IMDET can be reduced.

A computer program for causing a computer to implement the functions ofthe various units constituting the orthogonal transform apparatusaccording to the above embodiment or its modified example may bedistributed in the form stored in a semiconductor memory or in the formrecorded on a recording medium such as a magnetic recording medium or anoptical recording medium. Likewise, a computer program for causing acomputer to implement the functions of the various units constitutingthe audio decoding apparatus according to the above embodiment or itsmodified example may be distributed in the form stored in asemiconductor memory or in the form recorded on a recording medium suchas a magnetic recording medium or an optical recording medium. The term“recording medium” used here does not include a carrier wave.

FIG. 22 is a diagram illustrating the configuration of a computer thatoperates as the audio decoding apparatus by executing a computer programfor implementing the functions of the various units constituting theaudio decoding apparatus according to the above embodiment or itsmodified example.

The computer 100 includes a user interface unit 101, a communicationinterface unit 102, a storage unit 103, a storage media access device104, a processor 105, and an audio interface unit 106. The processor 105is connected to the user interface unit 101, communication interfaceunit 102, storage unit 103, storage media access device 104, and audiointerface unit 106, for example, via a bus.

The user interface unit 101 includes, for example, an input device suchas a keyboard and a mouse, and a display device such as a liquid crystaldisplay. Alternatively, the user interface unit 101 may include adevice, such as a touch panel display, into which an input device and adisplay device are integrated. The user interface unit 101 generates,for example, in response to a user operation, an operation signal forselecting audio data to be decoded, and supplies the operation signal tothe processor 105.

The communication interface unit 102 may include a communicationinterface for connecting the computer 100 to an audio data encodingapparatus, for example, a video camera, and a control circuit for thecommunication interface. Such a communication interface may be, forexample, a Universal Serial Bus (USB) interface.

Further, the communication interface unit 102 may include acommunication interface for connecting to a communication networkconforming to a communication standard such as the Ethernet (registeredtrademark), and a control circuit for the communication interface.

In the latter case, the communication interface unit 102 receivesencoded audio data to be decoded from another apparatus connected to thecommunication network, and passes the received data to the processor105.

The storage unit 103 includes, for example, a readable/writablesemiconductor memory and a read-only semiconductor memory. The storageunit 103 stores a computer program for implementing the audio decodingprocess to be executed on the processor 105, and also stores the datagenerated as a result of or during the execution of the program.

The storage media access device 104 is a device that accesses a storagemedium 108 such as a magnetic disk, a semiconductor memory card, or anoptical storage medium. The storage media access device 104 accesses thestorage medium 108 to read out, for example, the computer program foraudio decoding to be executed on the processor 105, and passes thereadout computer program to the processor 105.

The processor 105 decodes the encoded audio data by executing the audiodecoding computer program according to the above embodiment or itsmodified example. The processor 105 outputs the decoded audio data to aspeaker 107 via the audio interface unit 106.

The orthogonal transform apparatus according to the above embodiment orits modified example may be adapted for use in applications other thanthe decoding of the audio signals encoded in accordance with the MPEGSurround System. The orthogonal transform apparatus according to theabove embodiment or its modified example can be applied to various kindsof apparatus that need to transform MDCT coefficients into QMFcoefficients.

Further, the audio decoding apparatus according to the above embodimentor its modified example can be incorporated in various kinds ofapparatus, such as a computer, a video signal recording/reproductionmachine, etc., used to reproduce encoded audio signals.

All examples and conditional language recited herein are intended forpedagogical purposes to aid the reader in understanding the inventionand the concepts contributed by the inventor to furthering the art, andare to be construed as being without limitation to such specificallyrecited examples and conditions, nor does the organization of suchexamples in the specification relate to a showing of superiority andinferiority of the invention. Although the embodiments of the presentinvention have been described in detail, it should be understood thatthe various changes, substitutions, and alterations could be made heretowithout departing from the spirit and scope of the invention.

What is claimed is:
 1. An orthogonal transform apparatus fortransforming a plurality of modified discrete cosine transformcoefficients contained in a prescribed interval into a coefficientsequence containing a plurality of quadrature mirror filtercoefficients, comprising: an inverse exponential transform unit whichcomputes either one of the real and imaginary components of thequadrature mirror filter coefficient contained in a first subinterval ofa plurality subintervals among which the coefficient sequence is dividedso that the values of basis functions used to compute the coefficientsequence are symmetrically placed, by computing a sum of products of theplurality of modified discrete cosine transform coefficients and thebasis functions corresponding to the first subinterval, and computes theother one of the real and imaginary components of the quadrature mirrorfilter coefficient contained in the first subinterval and the real andimaginary components of the quadrature mirror filter coefficientcontained in another subintervals of the plurality of subintervals byperforming a butterfly operation using a computed value produced as aresult of the sum of products; and a coefficient adjusting unit whichcomputes the quadrature mirror filter coefficients by combining the realcomponent and the imaginary component for each of the plurality ofquadrature mirror filter coefficients.
 2. The orthogonal transformapparatus according to claim 1, wherein the inverse exponentialtransform unit comprises: a local multiplication unit which produces thecomputed value by computing the sum of products of the plurality ofmodified discrete cosine transform coefficients and the basis functionscorresponding to the first subinterval; a storage unit which stores thecomputed value; and a coefficient computing unit which retrieves thecomputed value from the storage unit and computes, for each of theplurality of subintervals, the real and imaginary components of thequadrature mirror filter coefficient contained in the subinterval. 3.The orthogonal transform apparatus according to claim 2, wherein thelocal multiplication unit produces the computed value by computing thesum of the products of odd-numbered modified discrete cosine transformcoefficients and the values of odd-numbered basis functions among thebasis functions corresponding to the real component of the quadraturemirror filter coefficient contained in the first subinterval and the sumof the products of even-numbered modified discrete cosine transformcoefficients and the values of even-numbered basis functions among thebasis functions corresponding to the real component of the quadraturemirror filter coefficient contained in the first subinterval.
 4. Theorthogonal transform apparatus according to claim 3, wherein byreversing the order of the real components of the quadrature mirrorfilter coefficients contained in the first subinterval, the coefficientcomputing unit computes the real components of the quadrature mirrorfilter coefficients contained in the another subintervals of theplurality of subintervals in which the values of the basis functions aresymmetrical in position to the values of the basis functionscorresponding to the first subinterval.
 5. The orthogonal transformapparatus according to claim 4, wherein the first subinterval containseither one of first and second halves of the first half of thecoefficient sequence and either one of first and second halves of thesecond half of the coefficient sequence.
 6. The orthogonal transformapparatus according to claim 5, wherein the coefficient computing unitcomputes the imaginary component of the quadrature mirror filtercoefficient contained in the first or second half of the first half ofthe coefficient sequence, whichever half is not contained in the firstsubinterval, by compensating for a phase difference between the basisfunctions corresponding to the real component of the quadrature mirrorfilter coefficient and the basis functions corresponding to theimaginary component of the quadrature mirror filter coefficient andthereby compensating the computed value calculated for the first orsecond half of the second half of the coefficient sequence, whicheverhalf is contained in the first subinterval.
 7. The orthogonal transformapparatus according to claim 1, further comprising: a second inverseexponential transform unit which computes the real components of theplurality of quadrature mirror filter coefficients by applying aninverse modified discrete cosine transform using a fast Fouriertransform to the plurality of modified discrete cosine transformcoefficients, and which computes the imaginary components of theplurality of quadrature mirror filter coefficients by applying aninverse modified discrete sine transform using a fast Fourier transformto the plurality of modified discrete cosine transform coefficients; anda switching unit which causes the inverse exponential transform unit orthe second inverse exponential, transform unit, whichever unit isselected according to the number of modified discrete cosine transformcoefficients contained in the prescribed interval, to compute the realand imaginary components of the plurality of quadrature mirror filtercoefficients.
 8. The orthogonal transform apparatus according to claim7, wherein when the number of modified discrete cosine transformcoefficients contained in the prescribed interval is smaller than 8, orwhen the number is not a power of 2, the switching unit causes theinverse exponential transform unit to compute the real and imaginarycomponents of the plurality of quadrature mirror filter coefficients, onthe other hand, when the number of modified discrete cosine transformcoefficients contained in the prescribed interval is equal to or largerthan 8, and when the number is a power of 2, the switching unit causesthe second inverse exponential transform unit to compute the real andimaginary components of the plurality of quadrature mirror filtercoefficients.
 9. The orthogonal transform apparatus according to claim7, wherein the second inverse exponential transform unit comprises: aninterchanging unit which interchanges the modified discrete cosinetransform coefficients contained in the first half of the prescribedinterval with the modified discrete cosine transform coefficientscontained in the second half of the prescribed interval; an invertingunit which inverts the sign of the modified discrete cosine transformcoefficients contained in the second half of the prescribed intervalafter the interchange; a first sub inverse cosine transform unit whichcomputes first coefficients by applying the inverse modified discretecosine transform using the fast Fourier transform to the modifieddiscrete cosine transform coefficients contained in the first half ofthe prescribed interval after the interchange; a second sub inversecosine transform unit which computes second coefficients by applying theinverse modified discrete cosine transform using the fast Fouriertransform to the sign-inverted modified discrete cosine transformcoefficients contained in the second half of the prescribed intervalafter the interchange; and an adder which computes the real componentsof the quadrature mirror filter coefficients by adding the first andsecond coefficients.
 10. An orthogonal transform method for transforminga plurality of modified discrete cosine transform coefficients containedin a prescribed interval into a coefficient sequence containing aplurality of quadrature mirror filter coefficients, comprising:computing either one of the real and imaginary components of thequadrature mirror filter coefficient contained in a first subinterval ofa plurality subintervals among which the coefficient sequence is dividedso that the values of basis functions used to compute the coefficientsequence are symmetrically placed, by computing a sum of products of theplurality of modified discrete cosine transform coefficients and thebasis functions corresponding to the first subinterval; computing theother one of the real and imaginary components of the quadrature mirrorfilter coefficient contained in the first subinterval and the real andimaginary components of the quadrature mirror filter coefficientcontained in another subintervals of the plurality of subintervals byperforming a butterfly operation using a computed value produced as aresult of the sum of products; and computing the quadrature mirrorfilter coefficients by combining the real component and the imaginarycomponent for each of the plurality of quadrature mirror filtercoefficients.
 11. The orthogonal transform method according to claim 10,wherein the computing either one of the real and imaginary components ofthe quadrature mirror filter coefficient contained in the firstsubinterval comprises: producing the computed value by computing the sumof products of the plurality of modified discrete cosine transformcoefficients and the basis functions corresponding to the firstsubinterval; storing the computed value; and retrieving the computedvalue from the storage unit and computing, for each of the plurality ofsubintervals, the real and imaginary components of the quadrature mirrorfilter coefficient contained in the subinterval.
 12. The orthogonaltransform method according to claim 11, wherein the producing thecomputed value produces the computed value by computing the sum of theproducts of odd-numbered modified discrete cosine transform coefficientsand the values of odd-numbered basis functions among the basis functionscorresponding to the real component of the quadrature mirror filtercoefficient contained in the first subinterval and the sum of theproducts of even-numbered modified discrete cosine transformcoefficients and the values of even-numbered basis functions among thebasis functions corresponding to the real component of the quadraturemirror filter coefficient contained in the first subinterval.
 13. Theorthogonal transform method according to claim 12, wherein by reversingthe order of the real components of the quadrature mirror filtercoefficients contained in the first subinterval, the computing the realand imaginary components of the quadrature mirror filter coefficientcontained in the subinterval computes the real components of thequadrature mirror filter coefficients contained in the anothersubintervals of the plurality of subintervals in which the values of thebasis functions are symmetrical in position to the values of the basisfunctions corresponding to the first subinterval.
 14. The orthogonaltransform method according to claim 13, wherein the first subintervalcontains either one of first and second halves of the first half of thecoefficient sequence and either one of first and second halves of thesecond half of the coefficient sequence.
 15. The orthogonal transformmethod according to claim 14, wherein the computing the real andimaginary components of the quadrature mirror filter coefficientcontained in the subinterval computes the imaginary component of thequadrature mirror filter coefficient contained in the first or secondhalf of the first half of the coefficient sequence, whichever half isnot contained in the first subinterval, by compensating for a phasedifference between the basis functions corresponding to the realcomponent of the quadrature mirror filter coefficient and the basisfunctions corresponding to the imaginary component of the quadraturemirror filter coefficient and thereby compensating the computed valuecalculated for the first or second half of the second half of thecoefficient sequence, whichever half is contained in the firstsubinterval.
 16. A non-transitory computer-readable recording mediumhaving recorded thereon an orthogonal transform computer program thatcauses a computer to execute a process of transforming a plurality ofmodified discrete cosine transform coefficients contained in aprescribed interval into a coefficient sequence containing a pluralityof quadrature mirror filter coefficients, the process comprising:computing either one of the real and imaginary components of thequadrature mirror filter coefficient contained in a first subinterval ofa plurality subintervals among which the coefficient sequence is dividedso that the values of basis functions used to compute the coefficientsequence are symmetrically placed, by computing a sum of products of theplurality of modified discrete cosine transform coefficients and thebasis functions corresponding to the first subinterval; computing theother one of the real and imaginary components of the quadrature mirrorfilter coefficient contained in the first subinterval and the real andimaginary components of the quadrature mirror filter coefficientcontained in another subintervals of the plurality of subintervals byperforming a butterfly operation using a computed value produced as aresult of the sum of products; and computing the quadrature mirrorfilter coefficients by combining the real component and the imaginarycomponent for each of the plurality of quadrature mirror filtercoefficients.
 17. An audio decoding apparatus for decoding amultichannel audio signal from a data stream containing a main signalcode into which a main signal representing a main component of eachchannel obtained by downmixing each channel signal of the multichannelaudio signal is encoded, a residual signal code into which coefficientsobtained by performing a modified discrete cosine transform on aresidual signal orthogonal to the main signal are encoded, and a spatialinformation code into which spatial information representing the degreeof interchannel similarity and interchannel intensity difference isencoded, the audio decoding apparatus comprising: a demultiplexing unitwhich demultiplexes the main signal code, the residual signal code, andthe spatial information code from the data stream; a main signaldecoding unit which reconstructs the main signal in a time domain bydecoding the main signal code; a quadrature mirror filtering unit whichtransforms the main signal in the time domain into quadrature mirrorfilter coefficients in a time-frequency domain by applying quadraturemirror filtering to the main signal; a spatial information decoding unitwhich reconstructs the spatial information by decoding the spatialinformation code; a residual signal decoding unit which reconstructs themodified discrete cosine transform coefficients of the residual signalby decoding the residual signal code; an orthogonal transform unitwhich, for each of a plurality of prescribed intervals generated so asto overlap each other by one half by dividing an entire frequency band,transforms the modified discrete cosine transform coefficients of theresidual signal contained in the prescribed interval into a coefficientsequence containing a plurality of quadrature mirror filter coefficientsin the time-frequency domain; an upmixing unit which computes quadraturemirror filter coefficients for each channel of the audio signal byupmixing the quadrature mirror filter coefficients of the main signaland the quadrature mirror filter coefficients of the residual signal byusing the spatial information; and an inverse quadrature mirrorfiltering unit which reconstructs each channel signal of the audiosignal by applying inverse quadrature mirror filtering to the quadraturemirror filter coefficients of each channel, and wherein the orthogonaltransform unit comprises: an inverse exponential transform unit whichcomputes either one of the real and imaginary components of thequadrature mirror filter coefficient of the residual signal contained ina first subinterval of a plurality subintervals among which thecoefficient sequence is divided so that the values of basis functionsused to compute the coefficient sequence are symmetrically placed, bycomputing a sum of products of the plurality of modified discrete cosinetransform coefficients and the basis functions corresponding to thefirst subinterval, and computes the other one of the real and imaginarycomponents of the quadrature mirror filter coefficient of the residualsignal contained in the first subinterval and the real and imaginarycomponents of the quadrature mirror filter coefficient of the residualsignal contained in another subintervals of the plurality ofsubintervals by performing a butterfly operation using a computed valueproduced as a result of the sum of products; and a coefficient adjustingunit which computes the quadrature mirror filter coefficients bycombining the real component and the imaginary component for each of theplurality of quadrature mirror filter coefficients of the residualsignal.